Strange shading on surface. Normals facing right way, no double faces, no double vertices
You have autosmooth enabled, you just need to disable it:



Since your objects have all triangular faces, what you see is the result of auto smooth shading which enhances those triangular shapes.

edit: to other readers benefit, I'll add that as (again) @Mr Zak pointed out, the real reason for this behavior should be "custom split normals" created by the external application from which it was imported this file in Blender.

below an image showing how to remove those custom split normal data (I couldn't find it before), and the effect on 3d viewport shading. After clearing that custom data, the "autosmooth" limit angle data is enabled again, although it is not needed at all in this case.